# Does the fluctuations-dissipation theorem hold in active matter for macroscopic physical quantities?

I am trying to understand how the fluctuation–dissipation theorem applies to active matter.

I simulated a system with active motors which may consume energy from the environment to move and exert force on fibers.

All the chemical reactions have both positive on and off rates.

Assume I define a macroscopic physical quantity $$x(t)$$:

1. Is detailed balance preserved when applying FDT to the macroscopic order parameter, $$x(t)$$?

2. Do the fluctuations in $$x(t)$$ around its mean value $$\langle x\rangle_0$$ correspond to the power spectrum $$S_x(\omega) = \langle \hat{x}(\omega)\hat{x}^*(\omega) \rangle$$?

3. Is it still true to say that FDT relates $$x$$ to the imaginary part of the Fourier transform $$\hat{\chi}(\omega)$$ of the susceptibility $$\chi(t)$$ by:

$$S_x(\omega) = \frac{2 k_\mathrm{B} T}{\omega} \mathrm{Im}\,\hat{\chi}(\omega)$$

1. Maybe the proper way to draw insight on such a system is to use detrended fluctuation analysis, is it right?
• I believe none of the above results hold in active matter. The detailed balance is not satisfied in general. You can still define fluctuations using the power spectrum if you wish, but then, these fluctuations only set a bound on the susceptibility. – stochastic Aug 6 '19 at 20:45
• This could be a starting point for some a search through the literature arxiv.org/abs/1610.06112 – Adam Aug 14 '19 at 18:45
• journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.258001 – 0x90 Aug 27 '19 at 15:02

## 1 Answer

In general, the fluctuation-dissipation theorem does not hold for active matter systems.

A way to see it, is that fluctuation dissipation theorems usually require the system to be at equilibrium or close to equilibrium, a condition that is broken for most active matter systems.

However, you have several works related to this question, in particular whatever generalized FDT could hold for some active matter systems, see this recent arxiv (https://arxiv.org/abs/1909.03726 ) for a general overview of the FDT (case of active matter system is treated in sec. III.D)